When tech industry executives gather at the White House Monday, brainstorming ways to modernize government will be on the agenda. But on display will be President Donald Trump’s evolving relationship with America’s corporate chieftains.

Some 300 business leaders have met with Mr. Trump since he took office promising the nation’s top executives a direct line to the Oval Office and a chance to shape economic policy. The discussions have helped the president project an image of CEO-in-chief as he awaits a major legislative...